Interpreting distance from hyperplane in SVM

1. SVMs do not estimate probabilities, so technically you can't make a statement about the probabilities of points belonging to classes given the output of an SVM. However, it is common to use the alpha output and the true labels of the training data to train a logistic regression classifier from which you can get well-calibrated probability estimates out of the SVM. Also, libsvm can compute these probabilities itself, look it up in its manual. It requires retraining the SVM.

2. The SVs essentially are the hyperplane in a way (the hyperplane is a linear combination of the SVs). This means that these are precisely the points where the classifier is in most uncertainty about the classification (so if you change the features values of the SVs by epsilon the classifier changes; this generally does not happen for every point). In you want to think about the alpha as a probability you need to think the opposite, and the "probability" is higher the further a point is from the hyperplane.

If you ask for my opinion you should really use the libSVM probability estimation.

Quote from "Sparse Probabilistic Classifiers" by Romain Hérault, Yves Grandvalet

"The scores returned by support vector machines are often used as a confidence measures in the classification of new examples. However, there is no theoretical argument sustaining this practice. Thus, when classification uncertainty has to be assessed, it is safer to resort to classifiers estimating conditional probabilities of class labels."

More details in the paper http://www.machinelearning.org/proceedings/icml2007/papers/233.pdf